ap physics 1 proportional reasoning

Question

\ An object accelerates from rest at a rate 2A from time t=0 to t=3T . it then accelerates at a new rate 4A from time t=3T to t=5T. What expression represents the displacement of object over time interval delta t = 5t?

how to do this question using proportional reasoning?

Answer 1

If you were comparing two displacements for v_i = 0, then you could use proportional thinking:

d = 1/2 a t² so d goes with a and t² . If 1st case is 1 and 2nd case is 2 then

d₂/d₁ = (a₂/a₁)(t₂/t₁)² would apply (but this analysis doesn't seem to apply)

However, I am reading this as d₁ = 1/2(2A)(3T)² = 9AT² and v_f1 = 2A(3T) = 6AT

d₂ = v_f1t₂ + 1/2 (a₂)(t₂)² or 6AT*2T + (1/2) (4A)(2T)² = 20AT²

d = d₁+d₂ = 29AT²